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Ellipses
An ellipse may be defined as a set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
These two points are said to be the foci of the ellipse and one point is called the focus of the ellipse.
The midpoint of the line segment joining both of the focus is called the centre of the ellipse. The line segment passing through the foci of the ellipse is called the major axis and the line segment passing through the centre and is perpendicular to the major axis is called the minor axis.
The endpoints of the major axes are said to be the vertices of the ellipse.
The geometric definition of an ellipse is the locus of all points such that the distances from two points called foci add up to a constant.
General equation
The general equation of an ellipse is [(x -h)a2 + (y- k)2/ b2 ] = 1 where the centre is (h ,k) and axes have lengths 2a and 2b.
The distance between the centre and either focus is c, where c2 = a2 - b2 where a > b > 0.
The standard equation of the ellipse with a vertical major axis is [(x - h)2/ a2 + (y- k)2/ b2 ] = 1 where the centre is (h ,k) and axes have lengths 2a and 2b.
The distance between the centre and either focus is c, where c2 = a2 - b2 where a > b > 0.
The eccentricity of an ellipse is e = c / a . For any ellipse 0 < e < 1.
Eccentricity is basically a measure of the ovalness of an ellipse. It is the ratio of the distance between the foci and the length of the major axis.
If the foci are very near the centre of the ellipse, the ellipse is nearly circular and e is close to zero.
If the foci are relatively far away from the centre, the ellipse is shaped more like an oval and e is closer to 1
Applications
The shape of a spotlight on a planar surface in most cases is an ellipse.
If you cut a cylinder at an angle you will get elliptical sections.
Lenses and mirrors can be elliptical in shape.
Some tanks are in fact elliptical in cross section. This gives them high capacity, but with a lower centre of gravity, so that they are more stable when being transported.
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